Optics of the Human Eye

Cover
Half Title
Series Information
Title Page
Copyright Page
Table of Contents
Acknowledgements
About the Authors
Sign Convention and Symbols
Introduction
Section I Basic Optical Structure of the Human Eye
1 The Human Eye: An Overview
1.1 Introduction
1.2 Optical Structure and Image Formation
1.3 The Retina
1.3.1 The Optic Disc and Blind Spot
1.4 The Cardinal Points
1.5 The Equivalent Power and Focal Lengths
1.6 Axes of the Eye
1.7 Center-Of-Rotation
1.8 Field-Of-Vision
1.9 Binocular Vision and Binocular Overlap
1.9.1 Interpupillary Distance
1.9.2 Binocular Overlap
1.10 Typical Dimensions
Summary of Main Symbols
References
2 Refracting Components: Cornea and Lens
2.1 Introduction
2.2 The Cornea
2.2.1 Anatomical Structure
2.2.2 Refractive Index
2.2.3 Radii of Curvature, Vertex Powers, and Total Corneal Power
2.2.4 Anterior Surface Shape
2.2.4.1 Toricity
2.2.4.2 Asphericity
2.2.5 Central Thickness
2.2.6 Posterior Surface Shape
2.2.7 Short-Term Changes in the Cornea
2.2.8 Positions of the Principal Points
2.3 The Lens and Its Parameters in the Unaccommodated State
2.3.1 Surface Radii of Curvature and Shapes
2.3.2 Thickness
2.3.3 Refractive Index Distribution
2.3.4 Equivalent Refractive Index
2.3.5 Equatorial Diameter
2.3.6 Lens Power
2.3.7 Positions of Principal Points
2.4 Accommodation
Summary of Main Symbols
References
3 The Pupil
3.1 Introduction – the Iris
3.2 Entrance and Exit Pupils
3.2.1 Effect of Aberrations
3.2.2 Accommodation
3.2.3 The Paraxial Marginal Ray and Paraxial Pupil Ray
3.3 Pupil Centration
3.4 Pupil Size
3.4.1 Level of Illumination
3.4.2 Field Size
3.4.3 Influences of Binocular Vision and Accommodation
3.4.4 Age
3.4.5 Drugs
3.4.6 Psychological Factors
3.4.7 Combining Factors
3.5 Pupil Aberration and Shape of the Obliquely Viewed Pupil
3.6 Significance of Pupil Size
3.6.1 Depth-Of-Field
3.6.2 Retinal Light Level
3.6.3 Retinal Image Quality and Visual Performance
3.6.4 Purpose of the Pupillary Light Response
3.7 Measurement of Pupil Size (Pupillometry)
3.8 Artificial Pupils
Summary of Main Symbols
References
4 Axes of the Eye
4.1 Introduction
4.2 Definitions and Significance
4.2.1 Optical Axis
4.2.2 Line of Sight
4.2.3 Visual Axis
4.2.4 Pupillary Axis
4.2.5 Fixation Axis
4.2.6 Keratometric Axis
4.2.7 Pupillary Circular Axis
4.3 Locating Some Axes
4.3.1 The Line of Sight
4.3.2 The Visual Axis
4.3.3 Keratometric Axis
4.4 Angles Between Axes
4.4.1 Visual Axis and Optical Axis: The Angle Alpha (α)
4.4.2 Pupillary Axis and Line of Sight: Angle Lambda (λ)
4.4.3 Pupillary Axis and the Visual Axis: Angle κ
4.4.4 Visual Axis and Achromatic Axis: Angle Psi (Ψ)
4.4.5 Fixation Axis and Optical Axis: Angle Gamma (γ)
Summary of Main Symbols
References
5 Paraxial Schematic Eyes
5.1 Introduction
5.2 Development of Paraxial Schematic Eyes
5.3 Gaussian Properties and Cardinal Points
5.3.1 Equivalent Power and Cardinal Points
5.3.2 The Aperture Stop and Entrance and Exit Pupils
5.3.3 Position and Magnification of Entrance Pupil
5.3.4 Paraxial Marginal Ray and Paraxial Pupil Ray
5.3.5 Paraxial Pupil Ray Angle Ratio
5.3.6 Effect of Accommodation
5.4 “Exact” Schematic Eyes
5.4.1 Gullstrand Number 1 (Exact) Eye
5.4.2 Le Grand Full Theoretical Eye
5.5 Simplified Schematic Eyes
5.5.1 Gullstrand Number 2 (Simplified) Eye as Modified By Emsley – the Gullstrand–Emsley Eye
5.5.2 Le Grand Simplified Eye
5.5.3 Bennett and Rabbetts’ Simplified Eye
5.6 Reduced Schematic Eyes
5.6.1 Emsley’s Reduced Eye (1952)
5.6.2 Bennett and Rabbetts (1989)
5.7 Variable Accommodating Eyes
5.7.1 Equivalent Power and Positions of Cardinal Points
Summary of Main Symbols
References
Section II Image Formation and Refraction
6 Image Formation: The Focused Paraxial Image
6.1 Introduction
6.2 The General Case
6.2.1 Retinal Image Size and Perceived Angular Size in Object Space
6.3 Eye Focused at Infinity
6.4 Binocular Vision
6.4.1 Stereopsis
6.4.2 Aniseikonia
Summary of Main Symbols
References
7 Refractive Anomalies
7.1 Introduction
7.2 Spherical Refractive Anomalies
7.2.1 Spherical Refractive Errors
7.2.1.1 Emmetropia (Normal Sight)
7.2.1.2 Myopia (Short Sight)
7.2.1.3 Hyperopia (Hypermetropia)
7.2.2 Presbyopia
7.3 Astigmatic Refractive Errors
7.4 Anisometropia
7.5 Distribution of Refractive Errors and Ocular Components
7.5.1 Distribution
7.5.2 Astigmatism
7.6 The Power of the Correcting Lens
7.6.1 Astigmatic Corrective Powers
7.6.2 Alternative System of Specifying Refractive Errors and Correcting Lens Powers
7.6.3 Thick Lenses and the Effect of Thickness
7.7 Effect of Parameter Changes On Refractive Errors
7.7.1 Refractive Error and Axial Length
7.7.2 Change in Corneal Curvature
7.7.3 Other Parameter Changes
Summary of Main Symbols
References
8 Measuring Refractive Errors
8.1 Introduction
8.2 Subjective Refraction Techniques
8.2.1 Simple Perception of Blur
8.2.1.1 Conventional Subjective Refraction Techniques
8.2.1.2 Optometers
8.2.2 Longitudinal Chromatic Aberration of the Eye
8.3 Subjective/Objective Refraction Techniques
8.3.1 Remote Refraction and Relay Systems
8.3.2 Split Image and Vernier Acuity (Coincidence Method)
8.3.3 Scheiner Principle
8.4 Objective Refraction Techniques
8.4.1 Retinoscopy
8.4.2 Parallax Movement Between Object and Image
8.4.3 Grating Focus
8.4.4 Image-Size Principle
8.4.5 Hartmann–Shack Aberrometer
8.4.6 Laser Raytracing Aberrometer
8.4.7 Photography
8.4.8 Visual Evoked Response
8.5 Accuracy and Reliability of Refraction
8.6 Factors Affecting Refraction
8.6.1 Target Factors
8.6.2 Optical Factors
8.6.3 Eccentric Viewing
8.6.4 Reference Plane
8.6.5 Site of Fundus Reflectance in Objective Refraction
8.6.6 Accommodation
8.6.7 Maximum Potential Visual Acuity
8.6.8 Discrepancies Between Subjective and Objective Refraction
Summary of Main Symbols
References
9 Image Formation: The Defocused Paraxial Image
9.1 Introduction
9.2 Retinal Image Size
9.2.1 The Size of the Defocused Image
9.2.2 An Eye Focused at a Finite Distance, Looking at an Object at Infinity
9.2.3 The Use of Artificial Pupils
9.3 Size of the Defocus Blur Disc
9.3.1 The Geometrical Aberration-Free Defocus Blur Disc
9.3.2 Experimentally Determined Angular Diameter of the Blur Discs
9.3.3 Defocus Ratio
9.4 Other Effects of Defocus
9.4.1 Alignment of Two Targets at Different Distances
9.4.2 Effect On Visual Acuity
9.4.2.1 The Value of K and the Corresponding Defocus Ratio
Summary of Main Symbols
References
10 Some Optical Effects of Ophthalmic Lenses
10.1 Introduction
10.2 Spectacle Magnification
10.3 Pupil Position and Magnification
10.3.1 Retinal Image Illuminance
10.4 Relative Spectacle Magnification
10.4.1 Axial Ametropia
10.4.2 Refractive Ametropia
10.4.3 Further Comments
10.5 Effects On Far and Near Points and Accommodative Demand
10.5.1 Accommodation Through a Correcting Lens
10.6 Rotation Magnification, Field-Of-View, and Field-Of-Vision
10.6.1 Rotational Magnification
10.6.2 Field-Of-View
10.6.3 Field-Of-Vision
10.7 Spectacle Lens Design
10.8 Contact Lens Optics
10.8.1 Over-Refraction
10.8.2 Fluid Lens
10.8.3 Spectacle Magnification
10.9 Intraocular Lens Optics
Summary of Main Symbols
References
Section III Light and the Eye
11 Light and the Eye: Introduction
11.1 Introduction
11.2 Radiation and the Electromagnetic Spectrum
11.3 Light
11.3.1 Photopic Vision
11.3.2 Mesopic Vision
11.3.3 Scotopic Vision
11.3.4 Photopic, Mesopic, and Scotopic Limits
11.4 Photometric Quantities, Units, and Example Levels
11.4.1 Luminous Flux (F)
11.4.2 Luminous Intensity (I)
11.4.3 Luminance (L)
11.4.4 Illuminance (E)
11.5 Some Useful Relationships
11.5.1 Luminous Intensity and Illuminance: The Inverse Square Law
11.5.2 Luminance and Illuminance
11.6 Which Photometric Quantity to Use
11.6.1 Threshold Detection
11.6.2 Supra-Threshold Visibility of Sources With a Small Angular Subtense
11.6.3 Supra-Threshold Visibility of Sources With a Large Angular Subtense
11.6.4 Measurement of Ambient Light Level
11.6.5 Other Comments
11.7 Colorimetry
Summary of Main Symbols
References
12 Passage of Light Into the Eye
12.1 Introduction
12.2 Specular Reflection
12.2.1 Images Formed By Multiple Reflections
12.3 Transmittance
12.3.1 Spectral Transmittance of the Whole Eye
12.3.2 Spectral Transmittance of Each Ocular Component
12.3.2.1 The Cornea
12.3.2.2 The Aqueous
12.3.2.3 The Lens
12.3.2.4 The Vitreous
12.3.3 Progressive Loss of Light as It Passes Through the Eye
12.3.4 Causes of Absorption Bands
12.3.5 Luminous Transmittance
12.4 Scatter
12.4.1 Scattering Theory
12.4.2 Cornea
12.4.3 Lens
12.4.4 Iris and Sclera
12.5 Fluorescence
12.6 Birefringence
12.6.1 Cornea
12.6.2 Lens
Summary of Main Symbols
References
13 Light Level at the Retina
13.1 Introduction
13.2 Retinal Illuminance: Directly Transmitted Light
13.2.1 On Axis
13.2.1.1 Large Area Sources
The Troland
Relationship Between Troland and Lux
13.2.1.2 The Point Source – Diffraction Limited
13.2.1.3 The Point Source – Aberrated
13.2.2 Off-Axis Or Peripheral Sources
13.3 Retinal Illuminance: Scattered Light
13.3.1 Effect of Position in the Lens of a Scattering Center
13.3.2 Measurement of Angular Distribution of Scattered Light
13.3.2.1 Conventional Threshold Method
13.3.2.2 Flicker Compensation Comparison Method
13.3.2.3 Hartmann–Shack Aberrometer
13.4 Photon Density Levels
13.4.1 Blackbody
13.4.2 Real Sources
13.5 Maxwellian View
13.5.1 Equivalent Luminance of a Lambertian Source
13.5.2 Adapting Pupil Size
Summary of Main Symbols
References
14 Light Interaction With the Fundus
14.1 Introduction
14.1.1 Inner Limiting Membrane to Photoreceptors (Six Layers)
14.1.2 The Photoreceptors
14.1.3 The Pigment Epithelium
14.1.4 The Choroid
14.1.5 The Sclera
14.2 Fundus Reflectance
14.2.1 Polarized Light
14.2.2 Guided and Unguided Light
14.2.3 Layers Responsible for the Fundus Reflectance
14.2.4 Veiling Glare
14.3 Absorption
14.4 Birefringence
14.5 The Stiles–Crawford Effect and Retinal Directionality
14.5.1 Peak of the Stiles–Crawford Effect
14.5.2 Integrating the Stiles–Crawford Effect Across the Pupil
14.5.3 Some Factors Influencing the Stiles–Crawford Effect
14.5.3.1 Wavelength
14.5.3.2 Eccentricity
14.5.3.3 Luminance
14.5.3.4 Field Size
14.5.3.5 Refractive Errors and Aberrations
14.5.3.6 Accommodation
14.5.3.7 Eye Disease
14.5.3.8 Phototropism
14.5.4 Theory
14.5.5 Measurement
14.5.6 Role of the Stiles–Crawford Effect
Summary of Main Symbols
References
Section IV Aberrations and Retinal Image Quality
15 Monochromatic Aberrations
15.1 Introduction
15.2 Representation of Monochromatic Aberrations
15.3 Specification of Monochromatic Aberrations – Taylor Series
15.4 Specification of Monochromatic Aberrations – Zernike Aberration System
15.4.1 Comparing Zernike and Taylor Aberration Systems
15.4.2 Right and Left Eyes
15.4.3 Using Zernike Aberrations to Determine Refraction
15.4.4 Zernike Aberrations Presented in Magnitude/Axis Format
15.4.5 Power Variation Across the Pupil
15.4.6 Other Aberration Systems
15.4.7 Reference Axis
15.4.8 Cornea and Lenticular Components of Aberrations
15.4.9 Change of Wavelength
15.4.10 Peripheral Aberrations
15.5 Techniques
15.6 Magnitudes of Aberrations
15.6.1 Central Field
15.6.2 Peripheral Field
15.6.2.1 Distortion
15.7 Influence of Monochromatic Aberrations On Visual Performance and Consequences of Correcting Them
Summary of Main Symbols
References
16 Monochromatic Aberrations of Optical Model Eyes
16.1 Introduction
16.2 Aberrations of Paraxial Schematic Eyes
16.2.1 On-Axis
16.2.2 Peripheral Field
16.2.3 Summary
16.3 Modeling Surface Shapes
16.3.1 Conicoid Surfaces
16.3.2 Figured Conicoid Surfaces
16.3.3 More Sophisticated Surfaces
16.3.4 Surface Types Used in Studies
16.4 Modeling the Lenticular Refractive Index Distribution
16.4.1 The Power of the Lens
16.4.1.1 Surface Powers
16.4.1.2 The Gradient Index Power
16.4.1.3 Total Lens Power and Positions of the Cardinal Points
16.5 Modeling the Retina
16.6 Survey of Finite Schematic Eyes
16.6.1 Lotmar (1971)
16.6.2 Drasdo and Fowler (1974)
16.6.3 Kooijman (1983)
16.6.4 Navarro Et Al. (1985)
16.6.5 Liou and Brennan (1997)
16.6.6 Reduced Eye Models of Thibos and Colleagues
16.7 Performance of Finite Schematic Eyes
16.7.1 On-Axis
16.7.2 Peripheral Field
16.7.3 Retinal Image Position
16.7.4 Retinal Illuminance
16.7.5 Summary
16.8 Other Optical Model Eyes, Including Customized Models
16.9 Which Eye Model to Use
Summary of Main Symbols
References
17 Chromatic Aberrations
17.1 Introduction
17.2 Longitudinal Chromatic Aberration
17.3 Transverse Chromatic Aberration
17.3.1 Chromatic Magnification
17.4 Measurement of Longitudinal Chromatic Aberration
17.4.1 Some Techniques
17.4.1.1 Best Focus Method
17.4.1.2 Vernier Method
17.4.1.3 Double-Pass Techniques
17.4.1.4 Chromo-Retinoscopy
17.4.1.5 Aberrometry
17.4.2 Magnitude
17.4.3 Wavelength in Focus
17.4.4 Effect of Accommodation and Refractive Error
17.5 Measurement of Transverse Chromatic Aberration
17.5.1 Technique
17.5.2 Magnitude
17.6 Effects of Chromatic Aberrations On Vision
17.6.1 Accommodation
17.6.2 Spatial Vision
17.6.3 Chromostereopsis
17.6.4 Aberrations of Ophthalmic Devices
17.7 Aberration Compensation and Correction
17.7.1 Natural Compensation Mechanism
17.7.2 Achromatizing Correcting Lenses
17.7.3 Other Compensation Methods
17.8 Modeling Chromatic Aberrations
17.8.1 Chromatic Dispersion
17.8.2 Schematic Eyes
17.8.2.1 Gaussian Properties
17.8.2.2 Chromatic Difference of Power and Chromatic Difference of Refraction
17.8.2.3 Chromatic Difference of Refraction of Reduced Schematic Eyes
17.8.2.4 Chromatic and Indiana Reduced Eyes of Thibos Et Al. (1992, 1997)
17.9 Estimating Lengths in Eyes
Summary of Main Symbols
References
18 Retinal Image Quality
18.1 Introduction
18.2 The Point and Line Spread Functions
18.2.1 The Diffraction-Limited PSF (Monochromatic Light)
18.2.2 The Aberrated PSF
18.2.3 The PSF and Its Use in Quantifying Image Quality
18.2.3.1 The Rayleigh Criterion (Diffraction-Limited and Monochromatic Sources)
18.2.3.2 The Half-Width
18.2.3.3 The Strehl Intensity Ratio
18.2.4 The PSF and LSF of Eyes
18.3 The Optical Transfer Function
18.3.1 The Diffraction-Limited OTF With No Stiles-Crawford Effect
18.3.2 Determination of the OTF
18.3.2.1 Determining the OTF From the Measured Wave Aberrations
18.3.2.2 Determining the OTF From the Aerial PSF
18.3.2.3 Determining the OTF From Psychophysical Comparison
18.3.3 OTF in the Presence of Defocus
18.3.3.1 The Geometrical Optical Approximation for Defocus
18.4 Retinal Image Quality
18.4.1 Central Vision
18.4.1.1 Defocus and Refraction
18.4.1.2 Polychromatic Light
18.4.1.3 The Stiles-Crawford Effect
18.4.1.4 Pupil Decentration
18.4.2 Peripheral Vision
18.5 More Image Quality Criteria – Metrics
Summary of Main Symbols
References
Section V Miscellaneous
19 Depth-Of-Field
19.1 Introduction
19.2 Criteria for Determining Depth-Of-Field
19.2.1 Criterion 1: The Range of Focusing Errors for Which No Perceptible Blur of a Target Is Noticeable
19.2.2 Criterion 2: The Range of Focusing Errors for Which the Visual Acuity Or Contrast Sensitivity Does Not Decrease Below a Particular Level Or By More Than a Certain Amount
19.2.3 Criterion 3: The Range of Focusing Errors for Which Changes in Contrast Are Not Detected for a Target in Longitudinal Sinusoidal Motion
19.2.4 Criterion 4: The Range of Focusing Errors for Which the Accommodation Response Does Not Change
19.2.5 Criterion 5: The Range of Focusing Errors Which Degrades Retinal Image Quality Below a Particular Level Or By More Than a Certain Amount
19.3 Modeling Depth-Of-Field
19.3.1 Criterion 1: The Range of Focusing Errors for Which No Perceptible Blur of a Target Is Noticeable
19.3.1.1 Effects of Diffraction and Aberrations
19.3.1.2 Influence of Diffraction Alone at Small Pupil Diameters
19.3.1.3 Influence of Aberrations Alone at Large Pupil Diameters
19.3.1.4 More Complex Objects
19.3.2 Criterion 5: The Range of Focusing Errors, Which Degrades Retinal Image Quality Below a Particular Level Or By More Than a Certain Amount
19.4 Methods for Increasing Depth-Of-Field
19.4.1 Artificial Pupils
19.4.2 Aspheric Surfaces
19.4.3 Other Methods
Summary of Main Symbols
References
20 The Aging Eye
20.1 Introduction
20.2 Cornea
20.2.1 Corneal Shape
20.2.2 Corneal Thickness
20.2.3 Transmittance
20.3 Anterior Chamber
20.4 Pupil Diameter
20.5 Lens
20.5.1 Shape
20.5.2 Thickness
20.5.3 Refractive Index Distribution
20.5.4 Equivalent Refractive Index
20.5.5 Diameter
20.5.6 Lens Power
20.5.7 Lenticular Transmittance, Scatter, and Fluorescence
20.6 Vitreous Chamber
20.7 Refractive Errors and Axial Length
20.8 Accommodation and Presbyopia
20.8.1 Presbyopia Theories
20.8.1.1 Lenticular Theories – Mechanical Changes in Lens and Capsule
20.8.1.2 Lenticular Theories – Geometric Theories
20.8.1.3 Extra-Lenticular Theories – Duane’s Theory and Changes in Ciliary Muscle
20.8.1.4 Extra-Lenticular Theories – Changes in Elastic Components of Zonules And/or Ciliary Body
20.8.1.5 Summary
20.9 Aberrations
20.10 Photometry
20.10.1 Effect of Light Loss On Visual Performance
20.10.2 Stiles–Crawford Effect
20.11 Schematic Eyes
Summary of Main Symbols
References
Appendices
Appendix 1 Paraxial Optics
A1.1 Introduction
A1.1.1 Finite Raytracing
A1.2 The Paraxial Approximations and Paraxial Rays
A1.2.1 Definition of a Paraxial Ray
A1.2.2 Paraxial Raytrace Equations
A1.3 A Paraxial Raytracing Scheme
A1.3.1 Step 1: Choosing a Ray
A1.3.2 Step 2: Refraction at the Jth Surface
A1.3.3 Step 3: Transfer to the Next (J + 1)th Surface
A1.3.4 Image Size and Magnification
A1.3.5 Special Case of the Object at Infinity
A1.3.6 Choice of Ray
A1.4 The Optical Invariant
A1.5 Cardinal Points and Equivalent Power
A1.6 The Lens Equation
A1.7 Gaussian Optics
Summary of Main Symbols
Appendix 2 Seidel Aberration Theory
A2.1 Quantification of Aberrations
A2.1.1 Rays From an Axial Point
A2.1.2 Rays From an Off-Axis Point
A2.2 The Wave Aberration Function
A2.2.1 Units of Aberrations
A2.2.2 Defocus and Wave Aberration
A2.2.3 Calculation of the Wave Aberration Function
A2.3 Seidel Aberrations
A2.3.1 Seidel Aberrations and the Primary Wave Aberration Coefficients
A2.3.2 Sagittal, Tangential, and Petzval Surfaces
A2.3.3 Wave Aberration Coefficients W2,0 and W2,2
A2.3.4 Modifications for a Curved Retina
A2.3.5 Seidel Aberrations of a Gradient Index Medium
Summary of Main Symbols
References
Appendix 3 Schematic Eyes
A3.1 Introduction
A3.1.1 Units
A3.2 Paraxial Schematic Eyes
A3.2.1 List of Eyes
A3.2.2 Relaxed “Exact” Schematic Eyes
A3.2.3 Relaxed Simplified Schematic Eyes
A3.2.4 Reduced Eyes
A3.2.5 Accommodated ‘Exact’ Schematic Eyes
A3.2.6 Accommodated Simplified Schematic Eyes
A3.3 Finite Schematic Eyes
A3.3.1 List of Eyes
Summary of Main Symbols
References
Appendix 4: Refraction Powers Across the Pupil
Summary of Main Symbols
Reference
Appendix 5 Calculation of PSF and OTF From Aberrations of an Optical System
A5.1 The Point Spread Function (PSF)
A5.1.1 The Wave Aberration Function W(X, Y)
A5.1.1.1 Defocus
A5.1.1.2 Chromatic Aberration
A5.1.1.3 Polychromatic Sources
A5.1.1.4 Computation Checks
A5.2 The Optical Transfer Function (OTF)
Summary of Main Symbols
References
Index